正则群（酉群和Weyl-Heisenberg群的半直积）在非对易扩展相空间上作为动力学群的表示
Representations of the Canonical group, (the semi-direct product of the Unitary and Weyl-Heisenberg groups), acting as a dynamical group on noncommuting extended phase space
摘要：
庞加莱群P的覆盖群的酉不可约表示定义了物理闵可夫斯基空间P/L上的粒子物理学的框架，其中L是洛伦兹群。虽然非常成功，但它没有提供足够大的对称群来包含具有SU（3）分类的观察到的粒子。玻恩提出了互易原理，即态物理在互易变换下必须是不变的，互易变换为{t，e，q，p}->{t，e，p，-q}，其中{t，e，q，p}是时间、能量、位置和动量的自由度。这意味着存在一个共轭的相对性原理，使得动量的变化率必须以b为界，其中b是一个普适常数。体现这一点的动力学对称性的适当群是正则群C（1，3）= U（1，3）*s H（1，3），并且在该理论中，非对易空间Q= C（1，3）/ SU（1，3）是被赋予作为正则群的第二Casimir不变量的度量的物理量子空间，T^2 + E^2 - Q^2/c^2-P^2/b^2 +（2 h I/bc）（Y/bc-2）其中{T，E，Q，P，I，Y}是Os（1，3）代数的生成元。我们的想法是使用麦基理论来研究正则动力学群的表示，以确定这些表示是否可以包含粒子态的谱。正则群的酉不可约表示包含一个直积项，它是卡尔曼作为强子动力学群研究的U（1，3）的表示。U（1，3）表示包含可以分解成无限阶梯的离散级数，其中梯级是对应于其余或空框架的U（3）（有限维）或C（2）（退化U（1）* SU（2）有限维表示）的表示。
英文摘要：
The unitary irreducible representations of the covering group of the Poincare group P define the framework for much of particle physics on the physical Minkowski space P/L, where L is the Lorentz group. While extraordinarily successful, it does not provide a large enough group of symmetries to encompass observed particles with a SU(3) classification. Born proposed the reciprocity principle that states physics must be invariant under the reciprocity transform that is heuristically {t,e,q,p}->{t,e,p,-q} where {t,e,q,p} are the time, energy, position, and momentum degrees of freedom. This implies that there is reciprocally conjugate relativity principle such that the rates of change of momentum must be bounded by b, where b is a universal constant. The appropriate group of dynamical symmetries that embodies this is the Canonical group C(1,3) = U(1,3) *s H(1,3) and in this theory the non-commuting space Q= C(1,3)/ SU(1,3) is the physical quantum space endowed with a metric that is the second Casimir invariant of the Canonical group, T^2 + E^2 - Q^2/c^2-P^2/b^2 +(2h I/bc)(Y/bc -2) where {T,E,Q,P,I,Y} are the generators of the algebra of Os(1,3). The idea is to study the representations of the Canonical dynamical group using Mackey's theory to determine whether the representations can encompass the spectrum of particle states. The unitary irreducible representations of the Canonical group contain a direct product term that is a representation of U(1,3) that Kalman has studied as a dynamical group for hadrons. The U(1,3) representations contain discrete series that may be decomposed into infinite ladders where the rungs are representations of U(3) (finite dimensional) or C(2) (with degenerate U(1)* SU(2) finite dimensional representations) corresponding to the rest or null frames.
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